| Title:
|
More on cardinal invariants of analytic $P$-ideals (English) |
| Author:
|
Farkas, Barnabás |
| Author:
|
Soukup, Lajos |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
50 |
| Issue:
|
2 |
| Year:
|
2009 |
| Pages:
|
281-295 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Given an ideal $\mathcal I$ on $\omega $ let $\mathfrak{a} (\mathcal I)$ ($\bar{\mathfrak{a}}(\mathcal I)$) be minimum of the cardinalities of infinite (uncountable) maximal $\mathcal I$-almost disjoint subsets of $[{\omega}]^{\omega}$. We show that $\mathfrak{a} (\mathcal I_h)>{\omega}$ if $\mathcal I_h$ is a summable ideal; but $\mathfrak{a} ({\mathcal Z_{\vec \mu }})= {\omega}$ for any tall density ideal $\mathcal Z_{\vec \mu }$ including the density zero ideal $\mathcal Z$. On the other hand, you have $\mathfrak{b}\le \bar{\mathfrak{a}}(\mathcal I)$ for any analytic $P$-ideal $\mathcal I$, and $\bar{\mathfrak{a}}(\mathcal Z_{\vec \mu })\le \mathfrak{a}$ for each density ideal $\mathcal Z_{\vec \mu }$. For each ideal $\mathcal I$ on $\omega $ denote $\mathfrak{b}_{\mathcal I}$ and $\mathfrak{d}_{\mathcal I}$ the unbounding and dominating numbers of $\langle \omega ^\omega , \le_{\mathcal I}\rangle $ where $f\le_{\mathcal I} g$ iff $\{n\in \omega :f(n)> g(n)\}\in \mathcal I$. We show that $\mathfrak{b}_{\mathcal I}= \mathfrak{b}$ and $\mathfrak{d}_{\mathcal I}= \mathfrak{d}$ for each analytic $P$-ideal $\mathcal I$. Given a Borel ideal $\mathcal I$ on $\omega $ we say that a poset $\mathbb P$ is {\em $\mathcal I$-bounding\/} iff $\forall\, x\in \mathcal I\cap V^{\mathbb P}$ $\exists\, y\in \mathcal I\cap V$ $x\subseteq y$. $\mathbb P$ is {\em $\mathcal I$-dominating\/} iff $\exists\, y\in \mathcal I\cap V^{\mathbb P}$ $\forall\, x\in \mathcal I\cap V$ $x\subseteq^* y$. For each analytic $P$-ideal $\mathcal I$ if a poset $\mathbb P$ has the Sacks property then $\mathbb P$ is $\mathcal I$-bounding; moreover if $\mathcal I$ is tall as well then the property $\mathcal I$-bounding/$\mathcal I$-dominating implies ${\omega}^{\omega}$-bounding/adding dominating reals, and the converses of these two implications are false. For the density zero ideal $\mathcal Z$ we can prove more: (i) a poset $\mathbb P$ is $\mathcal Z$-bounding iff it has the Sacks property, (ii) if $\mathbb P$ adds a slalom capturing all ground model reals then $\mathbb P$ is $\mathcal Z$-dominating. (English) |
| Keyword:
|
analytic $P$-ideals |
| Keyword:
|
cardinal invariants |
| Keyword:
|
forcing |
| MSC:
|
03E17 |
| MSC:
|
03E35 |
| idZBL:
|
Zbl 1212.03035 |
| idMR:
|
MR2537837 |
| . |
| Date available:
|
2009-08-18T12:25:16Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133434 |
| . |
| Reference:
|
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| . |
